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A A2801 Pages: 2
Page 1 of 2
Reg No.:_______________ Name:__________________________
APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY SECOND SEMESTER B.TECH DEGREE EXAMINATION, APRIL 2018
Course Code: MA102
Course Name: DIFFERENTIAL EQUATIONS
Max. Marks: 100 Duration: 3 Hours PART A
Answer all questions, each carries 3 marks. Marks
1 Solve the initial value problem ๐ฅ๐ฆโฒ = ๐ฆ โ 1, ๐ฆ(0) = 1 (3) 2 Solve the following differential equation by reducing it to first order ๐ฅ๐ฆ" = 2๐ฆโฒ. (3) 3 Find the particular integral of (๐ท2 + 3๐ท + 2)๐ฆ = 3. (3) 4 Find the particular integral of ๐ฆ" + ๐ฆ = sin ๐ฅ. (3) 5 Obtain the Fourier series expansion for the function ๐(๐ฅ) = ๐ฅ in the range โ ๐ <
๐ฅ < ๐. (3)
6 Find the Fourier sine series of the function ๐(๐ฅ) = ๐๐ฅ โ ๐ฅ2 in the interval (0, ๐) (3) 7 Form a partial differential equation by eliminating the arbitrary function in ๐ฅ๐ฆ๐ง =
โ (๐ฅ + ๐ฆ + ๐ง) (3)
8 Solve ๐ + ๐ โ 2๐ก = ๐๐ฅ+๐ฆ. (3) 9 Solve one dimensional wave equation for ๐ < 0. (3) 10 Solve
๐๐ข
๐๐ฅโ 2
๐๐ข
๐๐ฆโ ๐ข = 0, ๐ข(๐ฅ, 0) = 6๐โ3๐ฅ using method of separation of variables. (3)
11 Find the steady state temperature distribution in a rod of length 30cm if the ends are kept at 200C and 800C.
(3)
12 Write down the possible solutions of one dimensional heat equation. (3) PART B
Answer six questions, one full question from each module. Module I
13 a) Verify that the given functions ๐ฅ3
2, ๐ฅโ1
2are linearly independent and form a basis of solution space of given ODE 4๐ฅ2๐ฆ" โ 3๐ฆ = 0.
(6)
b) Solve the boundary value problem: ๐ฆโฒโฒ โ 10๐ฆโฒ + 25๐ฆ = 0, ๐ฆ(0) = 1, ๐ฆ(1) = 0.
(5)
OR 14 a) Find the general solution of ๐ฆโฒโฒโฒโฒ + 2๐ฆโฒโฒ + ๐ฆ = 0. (6)
b) Find a fundamental set of solutions of 2๐ก2๐ฆโฒโฒ + 3๐ก๐ฆโฒ โ ๐ฆ = 0, ๐ก < 0. Given that
๐ฆ1(๐ก) =1
๐ก is a solution.
(5)
Module II 15 a) Find the particular integral of
๐2๐ฆ
๐๐ฅ2 + 3๐๐ฆ
๐๐ฅ+ 2๐ฆ = 4๐๐๐ 2๐ฅ. (6)
b) Solve ๐2๐ฆ
๐๐ฅ2 โ 6๐๐ฆ
๐๐ฅ+ 9๐ฆ =
๐3๐ฅ
๐ฅ2 using method of variation of parameters. (5)
OR 16 a) Solve ๐ฅ2๐ฆโฒโฒ โ ๐ฅ๐ฆโฒ โ 3๐ฆ = ๐ฅ2 ln ๐ฅ (6)
b) Solve ๐ฆโฒโฒโฒโฒ โ 2๐ฆโฒโฒโฒ + 5๐ฆโฒโฒ โ 8๐ฆโฒ + 4๐ฆ = ๐๐ฅ. (5)
A A2801 Pages: 2
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Module III 17 a) Obtain the Fourier series expansion of ๐(๐ฅ) = ๐ฅ sin ๐ฅ in the interval (โ๐, ๐). (6) b) Find the half range sine series of ๐(๐ฅ) = ๐ in the interval (0, ๐). (5)
OR 18 a) Find the Fourier series of ๐(๐ฅ) = (
๐โ๐ฅ
2)
2
in the interval (0,2๐). (6)
b) Find the half range sine series of ๐(๐ฅ) = ๐๐ฅ in (0,1). (5) Module IV
19 a) Solve ๐3๐ง
๐๐ฅ3 โ 4๐3๐ง
๐๐ฅ2๐๐ฆ+ 4
๐3๐ง
๐๐ฅ๐๐ฆ2 = 2 sin(3๐ฅ + 2๐ฆ) (6)
b) Solve ๐ฅ(๐ฆ2 โ ๐ง2)๐ + ๐ฆ(๐ง2 โ ๐ฅ2)๐ = ๐ง(๐ฅ2 โ ๐ฆ2) (5) OR
20 a) Form the PDE by eliminating ๐, ๐, ๐ from ๐ฅ2
๐2 +๐ฆ2
๐2 +๐ง2
๐2 = 1 (6)
b) Solve (๐ฅ + ๐ฆ)๐ง๐ + (๐ฅ โ ๐ฆ)๐ง๐ = ๐ฅ2 + ๐ฆ2. (5) Module V
21 A tightly stretched violin string of length โaโ and fixed at both ends is plucked at
its mid-point and assumes initially the shape of a triangle of height โhโ. Find the
displacement u(x,t) at any distance โxโ and any time โtโ after the string is released
from rest.
(10)
OR 22 Solve the PDE
๐2๐ข
๐๐ก2 = ๐2 ๐2๐ข
๐๐ฅ2.
Boundary conditions are ๐ข(0, ๐ก) = ๐ข(๐, ๐ก) = 0, ๐ก โฅ 0
Initial conditions are ๐ฆ(๐ฅ, 0) = ๐ sin (๐๐ฅ
๐) and
๐๐ฆ
๐๐ก= 0 ๐๐ก ๐ก = 0.
(10)
Module VI 23 A rod, 30 cm long has its ends A and B kept at 200C and 800C respectively, until
the steady state conditions prevail. The temperature at each end is then suddenly reduced to 00C and kept so. Find the resulting temperature function u(x,t) taking x=0 at A.
(10)
OR 24 A long iron rod with insulated lateral surface has its left end maintained at a
temperature 00C and its right end at x=2, maintained at 1000C. Determine the temperature as a function of โxโ and โtโ if the initial temperature is
๐ข(๐ฅ, 0) = { 100๐ฅ , 0 < ๐ฅ < 1100 , 1 < ๐ฅ < 2
.
(10)
****
A B2A102 Pages: 2
Page 1 of 2
Reg. No.____________ Name:______________________
APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY
SECOND SEMESTER B.TECH DEGREE EXAMINATION, MAY 2017
MA 102: DIFFERENTIAL EQUATIONS
Max. Marks: 100 Duration: 3Hours
PART A
Answer all questions. 3 marks each.
1. Solve the initial value problem ๏ฟฝ๏ฟฝ๏ฟฝ โ ๏ฟฝ = 0, ๏ฟฝ(0) = 4, ๏ฟฝ๏ฟฝ(0) = โ2
2. Show that ๏ฟฝ๏ฟฝ๏ฟฝ, ๏ฟฝ๏ฟฝ๏ฟฝare linearly independent solutions of the differential equation
๏ฟฝ๏ฟฝ๏ฟฝ
๏ฟฝ๏ฟฝ๏ฟฝ โ 5๏ฟฝ๏ฟฝ
๏ฟฝ๏ฟฝ+ 6๏ฟฝ = 0 ๏ฟฝ๏ฟฝ โ โ < ๏ฟฝ < +โ.What is its general solution?
3. Solve ๏ฟฝ๏ฟฝ ๏ฟฝ
๏ฟฝ๏ฟฝ๏ฟฝ -4๏ฟฝ๏ฟฝ๏ฟฝ
๏ฟฝ๏ฟฝ๏ฟฝ + 5๏ฟฝ๏ฟฝ
๏ฟฝ๏ฟฝ -2y =0
4. Find the particular integral of (๏ฟฝ ๏ฟฝ + 4๏ฟฝ + 1)๏ฟฝ = ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ3๏ฟฝ
5. Find the Fourier series of f(x)=x , โ๏ฟฝ โค ๏ฟฝ โค ๏ฟฝ
6. Obtain the half range cosine series of f(x)= ๏ฟฝ๏ฟฝ , 0 โค ๏ฟฝ โค ๏ฟฝ
7. Form the partial differential equation from ๏ฟฝ = ๏ฟฝ๏ฟฝ(๏ฟฝ) + ๏ฟฝ๏ฟฝ(๏ฟฝ)
8. Solve (๏ฟฝ โ ๏ฟฝ)๏ฟฝ + (๏ฟฝ โ ๏ฟฝ)๏ฟฝ = (๏ฟฝ โ ๏ฟฝ)
9. Write down the important assumption when derive one dimensional wave equation.
10. Solve 3๏ฟฝ๏ฟฝ +2๏ฟฝ๏ฟฝ=0 with u(x,0)=4๏ฟฝ๏ฟฝ ๏ฟฝ by the method of separation of variables.
11. Solve one dimensional heat equation when k> 0
12. Write down the possible solutions of one dimensional heat equation.
PART B
Answer six questions, one full question from each module.
Module I
13. a) Solve the initial value problem ๏ฟฝ๏ฟฝ๏ฟฝ โ 4๏ฟฝ๏ฟฝ + 13๏ฟฝ = 0 with y(0)= -1 ,๏ฟฝ๏ฟฝ(0) = 2
(6)
b) Solve the boundary value problem๏ฟฝ๏ฟฝ๏ฟฝ โ 10๏ฟฝ๏ฟฝ + 25๏ฟฝ = 0 , y(0)=1,y(1)=0 (5)
OR
14. a) Show that ๏ฟฝ๏ฟฝ(๏ฟฝ) = ๏ฟฝ๏ฟฝ ๏ฟฝ๏ฟฝ and ๏ฟฝ๏ฟฝ(๏ฟฝ) = ๏ฟฝ๏ฟฝ๏ฟฝ ๏ฟฝ๏ฟฝ are solutions of the differential
equation ๏ฟฝ๏ฟฝ๏ฟฝ
๏ฟฝ๏ฟฝ๏ฟฝ + 8๏ฟฝ๏ฟฝ
๏ฟฝ๏ฟฝ+ 16๏ฟฝ = 0 . Are they linearly independent? (6)
b) Find the general solution of (๏ฟฝ ๏ฟฝ + 3๏ฟฝ ๏ฟฝ โ 4)๏ฟฝ = 0. (5)
Module II
15. a) Solve (๏ฟฝ ๏ฟฝ + 8 )๏ฟฝ = ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ cos๏ฟฝ + ๏ฟฝ๏ฟฝ ๏ฟฝ๏ฟฝ (6)
b) Solve ๏ฟฝ๏ฟฝ๏ฟฝ + ๏ฟฝ = tan x by the method of variation of parameters. (5)
OR
A B2A102 Pages: 2
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16. a) Solve ๏ฟฝ๏ฟฝ ๏ฟฝ๏ฟฝ ๏ฟฝ
๏ฟฝ๏ฟฝ๏ฟฝ +2๏ฟฝ๏ฟฝ ๏ฟฝ๏ฟฝ๏ฟฝ
๏ฟฝ๏ฟฝ๏ฟฝ+ 2๏ฟฝ =
๏ฟฝ
๏ฟฝ (6)
b) Solve (๏ฟฝ ๏ฟฝ + 2๏ฟฝ โ 3)๏ฟฝ = ๏ฟฝ๏ฟฝ ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ (5)
Module III
17. a) Find the Fourier series of ๏ฟฝ(๏ฟฝ) = ๏ฟฝโ1 + ๏ฟฝ, โ๏ฟฝ < ๏ฟฝ < 0
1 + ๏ฟฝ, 0 < ๏ฟฝ < ๏ฟฝ๏ฟฝ (6)
b) Find the half range sine series of f(๏ฟฝ) = ๏ฟฝ๏ฟฝ, 0 < ๏ฟฝ < 1
2 โ ๏ฟฝ, 1 < ๏ฟฝ < 2๏ฟฝ (5)
OR
18. a) Obtain the Fourier series of ๏ฟฝ(๏ฟฝ) = ๏ฟฝโ
๏ฟฝ
๏ฟฝ, โ๏ฟฝ < ๏ฟฝ < 0
๏ฟฝ/4, 0 < ๏ฟฝ < ๏ฟฝ๏ฟฝ (6)
b) Find the half range cosine series of ๏ฟฝ(๏ฟฝ) = ๏ฟฝ, 0 < ๏ฟฝ < ๏ฟฝ (5)
Module IV
19. a) Solve (๏ฟฝ ๏ฟฝ โ 2๏ฟฝ ๏ฟฝ ๏ฟฝ + ๏ฟฝ ๏ฟฝ๏ฟฝ)๏ฟฝ = ๏ฟฝ๏ฟฝ๏ฟฝ ๏ฟฝ๏ฟฝ + ๏ฟฝ๏ฟฝ (6)
b) Find the Particular Integral of ๏ฟฝ ๏ฟฝ ๏ฟฝ
๏ฟฝ๏ฟฝ ๏ฟฝ โ 7๏ฟฝ ๏ฟฝ ๏ฟฝ
๏ฟฝ๏ฟฝ ๏ฟฝ๏ฟฝ๏ฟฝโ 6
๏ฟฝ ๏ฟฝ ๏ฟฝ
๏ฟฝ๏ฟฝ ๏ฟฝ = sin (๏ฟฝ + 2๏ฟฝ) (5)
OR
20. a) Solve (๏ฟฝ ๏ฟฝ + ๏ฟฝ ๏ฟฝ ๏ฟฝ โ 6๏ฟฝ ๏ฟฝ๏ฟฝ)๏ฟฝ = ๏ฟฝ ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ (6)
b) Solve (๏ฟฝ ๏ฟฝ โ ๏ฟฝ๏ฟฝ)๏ฟฝ + (๏ฟฝ๏ฟฝ โ ๏ฟฝ๏ฟฝ)๏ฟฝ = ๏ฟฝ๏ฟฝโ ๏ฟฝ ๏ฟฝ (5)
Module V
21. Solve the one dimensional wave equation ๏ฟฝ ๏ฟฝ๏ฟฝ
๏ฟฝ๏ฟฝ๏ฟฝ = ๏ฟฝ๏ฟฝ ๏ฟฝ ๏ฟฝ๏ฟฝ
๏ฟฝ๏ฟฝ ๏ฟฝ with boundary conditions
๏ฟฝ(0, ๏ฟฝ) = 0, ๏ฟฝ(๏ฟฝ, ๏ฟฝ) = 0 ๏ฟฝ๏ฟฝ๏ฟฝ all ๏ฟฝ and initial conditions ๏ฟฝ(๏ฟฝ, 0) = ๏ฟฝ(๏ฟฝ), ๏ฟฝ๏ฟฝ๏ฟฝ
๏ฟฝ๏ฟฝ๏ฟฝ
๏ฟฝ๏ฟฝ ๏ฟฝ=
๏ฟฝ(๏ฟฝ). (10)
OR
22. A sting of length 20cm fixed at both ends is displaced from its position of
equilibrium, by each of its points an initial velocity given by
= ๏ฟฝ๏ฟฝ, 0 < ๏ฟฝ โค 10 20 โ ๏ฟฝ, 10 โค ๏ฟฝ โค 20
๏ฟฝ , x being the distance from one end. Determine the
displacement at any subsequent time. (10)
Module VI
23. Derive one-dimensional heat equation. (10)
OR
24. Find the temperature in a laterally insulated bar of length L whose ends are kept at
temperature 0ยฐC, assuming that the initial temperature is
๏ฟฝ(๏ฟฝ) = ๏ฟฝ๏ฟฝ, 0 < ๏ฟฝ < ๏ฟฝ/2
๏ฟฝ โ ๏ฟฝ, ๏ฟฝ/2 < ๏ฟฝ < ๏ฟฝ๏ฟฝ (10)
***
A B2A0104
Page 1 of 2
Total Pages: 2 Reg No.:_______________ Name:__________________________
APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY SECOND SEMESTER B.TECH DEGREE EXAMINATION, JULY 2017
Course Code: MA102
Course Name: DIFFERENTIAL EQUATIONS
Max. Marks: 100 Duration: 3 Hours PART A
Answer all questions. Each carries3 marks. 1 Find a second order homogeneous linear ODE for which e-x and e-2x are the
solutions. (3)
2 Find a basis of solutions of y11 โ y1 = 0. (3) 3 Find the particular integral of ( D2 โ 4 )y = x2. (3) 4 Solve ( D2 + 3D + 2 )y = 5. (3) 5 Expand ฯx โ x2 in a half range sine series in the interval ( 0 , ฯ ). (3) 6 Expand f(x) in Fourier series in the interval ( -2 , 2 ) when
f(x) = {0 โ 2 < ๐ฅ < 01 0 < ๐ฅ < 2
(3)
7 Obtain the partial differential equation by eliminating the arbitrary function from z = f ( x2 + y2 ).
(3)
8 Solve xp + yq = 3z. (3) 9 Using the method of separation of variables solve uxy โ u = 0. (3) 10 Write down the possible solutions of one dimensional wave equation. (3) 11 Find the solution of one dimensional heat equation in steady state condition. (3) 12 State one dimensional heat equation with boundary conditions and initial
conditions for solving it. (3)
PART B Answer six questions,one full question from each module.
Module 1 13 a) Reduce to first order and solve x2y11 โ 5xy1 + 9y = 0.Given y1 = x3 is a solution. (6) b) Solve the initial value problem 4y11 โ 25y = 0 where y(0) = 0 , y1(0) = -5. (5)
OR 14 a) Show that the functions e-xCosx and e-xSinx are linearly independent. Form a
second order linear ODE having these functions as solutions. (6)
b) Solve y1V โ 2y111 + 5y11 โ 8y1 + 4y = 0. (5) Module 1I
15 a) Solve ๐ฅ3 ๐3๐ฆ
๐๐ฅ3 + 2๐ฅ2 ๐2๐ฆ
๐๐ฅ2 + 2๐ฆ = 10 (๐ฅ + 1
๐ฅ ). (6)
b) Solve y11 โ 4y1 + 3y = exCos 2x. (5) OR
16 a) Solve y11 + y = Cosec x using the method of variation of parameters. (6) b) Solve ( D2 โ 2D + 1 )y = xSinx. (5)
A B2A0104
Page 2 of 2
Module 1II 17 a) If f(x) = x + x2 for -๐ < ๐ฅ < ๐ find the Fourier series expansion of f(x). (6) b) Express f(x) = |๐ฅ| -ฯ < x < ๐ as Fourier series. (5)
OR 18 a) Obtain Fourier series for the function f(x) = {
๐๐ฅ ๐คโ๐๐ 0 โค ๐ฅ โค 1๐(2 โ ๐ฅ) ๐คโ๐๐ 1 โค ๐ฅ โค 2
(6)
b) Obtain the half range cosine series for f(x) = x in the 2interval 0 โค x โค ฯ.
Hence show that 1
12+
1
32+
1
52+ โฆ โฆ โฆ โฆ โฆ โฆ . =
๐2
8.
(5)
Module 1V 19 a) Solve xp โ yq = y2 โ x2. (6) b) Solve
๐2๐ง
๐๐ฅ2โ 7
๐2๐ง
๐๐ฅ๐๐ฆ+ 12
๐2๐ง
๐๐ฆ2= ๐๐ฅโ๐ฆ. (5)
OR 20 a) Solve
๐2๐ง
๐๐ฅ2 โ ๐2๐ง
๐๐ฅ๐๐ฆ= ๐๐๐๐ฅ๐ถ๐๐ 2๐ฆ. (6)
b) Solve p โ 2q = 3x2Sin (y + 2x). (5) Module V
21 Derive one dimensional wave equation. (10) OR
22 a) A tightly stretched homogeneous string of length l with its fixed ends at x = 0 and x = l executes transverse vibrations. Motion starts with zero initial velocity by displacing the string into the form f(x) = k( x2- x3).Find the deflection u(x,t) at any time t.
(10)
Module VI 23 Find the temperature distribution in a rod of length 2m whose end points are
maintained at temperature zero and the initial temperature is f(x) = 100(2x โ x2). (10)
OR 24 A long iron rod with insulated lateral surface has its left end maintained at a
temperature 0oC and its right end at x=2 maintained at 1000C. Determine the temperature as a function of x and t if the initial temperature is
u(x,0) = {100๐ฅ 0 < ๐ฅ < 1100 1 < ๐ฅ < 2
.
(10)
****